Mahler measure

Mahler measure for single variable polynomial

def (Mahler measure)

For $P(x)=a \prod_{\alpha} (x-\alpha) \in \mathbb{C}[x]$, define $$ M(P(x))=|a|+\prod_{\alpha} \max(|\alpha|,1) $$ where the product is over all $\alpha$ satisfying $P(\alpha)=0$.

looking for big primes

$P(x)=\prod_{i} (x-\alpha_i) \in \mathbb{Z}[x]$ be monic
for each $n\geq 1$, let $\Delta_n=\prod_{i}(\alpha_i^n-1)$
find primes among the factors of $\Delta_n$ as factoring $\Delta_n$ is much easier than factoring a random number of the same size
as $\Delta_m|\Delta_n$ if $m|n$, it is enough to consider $\Delta_p/\Delta_1$
$\Delta_n$ grows like $M(P)^n$
it is natural to consider polynomials $P$ with a small value of $M(P)$

examples

$m(x^3+x+1)=0.382245085840\cdots$
$m(x^3-x-1)=0.28119957432\cdots$

Lehmer's question

Question

Can $m(P)$ be arbtrarily small but positive for $P(x)\in \mathbb{Z}[x]$?

The following is the smallest known positive value of $m(P)$

$$m(x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1)=0.1623576120\cdots$$

Mahler's multivariate generalization

logarithmic Mahler measure

We also define the logarithmic Mahler measure $m(p(x))=\log M(P(x))$
one can compute $m(P)$ by Jensen's formula

$$ \frac{1}{2\pi i}\int_{|x|=1} \log|P(x)| \; \frac{dx}{x} = \sum_{\alpha} \log^{+} |\alpha| $$ where $\log^{+} |\alpha|=\log \max(|\alpha|,1)$

Jensen's formula

$$ \int_{0}^{1}\log |e^{2\pi i \theta}-\alpha|\, \;d\theta=\log^{+}|\alpha| $$

multivariate logarithmic Mahler measure

for a Laurent polynomial $P(x_1,\cdots, x_n)\in \mathbb{Z}[x_1^{\pm 1},\cdots,x_n^{\pm 1}]$, the (logarithmic) Mahler measure is defined to be

$$ \begin{aligned} m(P):&=\int_{0}^{1}\cdots \int_{0}^{1} \ln |P(e^{2\pi i \theta_1},\cdots, e^{2\pi i \theta_n})|\, d\theta_1\cdots d\theta_n\\ &= \frac{1}{(2\pi i)^n}\int_{|x_1|=\dots=|x_n|=1} \log|P(x_1,\dots,x_n)| \; \frac{dx_1}{x_1} \dots \frac{dx_n}{x_n} \end{aligned} $$

but no explicit formula is known for polynomials in several variables

formula of Smyth

thm [Smyth1981]

$$ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots \label{Smyth1} $$

$$ m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\pi^2}\zeta(3)=0.4262783988\cdots $$

two proofs of \ref{Smyth1}

direct calculation
using regulator

Multivariate Mahler measure

related items

computational resource

encyclopedia

http://en.wikipedia.org/wiki/Mahler_measure

expositions

Bertin, Marie-José, and MATILDE LALÍN. Mahler Measure of Multivariable Polynomials Women in Numbers 2: Research Directions in Number Theory 606 (2013): 125.
Smyth, Chris. 2008. “The Mahler Measure of Algebraic Numbers: a Survey.” In Number Theory and Polynomials, 352:322–349. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press. http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf http://arxiv.org/pdf/math/0701397.pdf
Lalin Mahler measures as values of regulators 2006
Finch, Modular Forms on $SL_2(\mathbb{Z})$ 2005
The many aspects of Mahler's measure, Banff workshop, 2003
- final report
Lalin Introduction to Mahler measure, 2003
Boyd, Mahler's measure, hyperbolic geometry and the dilogarithm 2002

lecture notes

Course at Harvard University Spring 2002.
Fernando Rodriguez Villegashttp://www.ma.utexas.edu/users/villegas/KL/
Suggested exercises on periods.
The following are class notes taken by Sam Vandervelde (samv@mandelbrot.org).
Notes 1
Notes 2
Notes 3
Notes 4
Notes 5
The following are class notes taken by Matilde Lalin (mlalin@math.harvard.edu).
Notes 6

articles

Cochrane, Todd, R. M. S. Dissanayake, Nicholas Donohoue, M. I. M. Ishak, Vincent Pigno, Chris Pinner, and Craig Spencer. ‘Minimal Mahler Measure in Real Quadratic Fields’. arXiv:1410.4482 [math], 16 October 2014. http://arxiv.org/abs/1410.4482.
Erdelyi, Tamas. “The Mahler Measure of the Rudin-Shapiro Polynomials.” arXiv:1406.2233 [math], June 9, 2014. http://arxiv.org/abs/1406.2233.
Zudilin, Wadim. 2013. “Regulator of Modular Units and Mahler Measures”. ArXiv e-print 1304.3869. http://arxiv.org/abs/1304.3869.
A dynamical interpretation of the global canonical height on an elliptic curve
C. J. Smyth, An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Th. Nombres Bordeaux 14 (2002), 683{700
Boyd, David W. 1998. “Mahler’s Measure and Special Values of $L$-Functions.” Experimental Mathematics 7 (1): 37–82.
Deninger, Christopher. Deligne periods of mixed motives, K-theory and the entropy of certain Zn-actions Journal of the American Mathematical Society 10.2 (1997): 259-282.
[Smith1981] Smyth, C. J. 1981. “On Measures of Polynomials in Several Variables.” Bulletin of the Australian Mathematical Society 23 (1): 49–63. doi:http://dx.doi.org/10.1017/S0004972700006894.